Tuesday, July 12, 2016

Universality in mathematics

Physicists tend to think of universality (the details don't matter) as a physical phenomena. However, in the early days of this blog I posted about how I became aware that the central limit theorem in mathematics is an example of universality.

"A histogram of the first two digits of accounts payable data of a major software company, together with the Benford's law prediction. (Source: Journal of Accountancy)"

The nicest part of the article is the discussion of the connections between random matrix theory, energy level spacings in nuclear physics, and the spacing of prime numbers and zeros of the Riemann zeta function. Universality is reflected in them all being described by the same probability distribution.

"A histogram of spacings between bus arrival times in Cuernavaca (Mexico), in crosses; the solid line is the prediction from random matrix theory. (Source: Krbalek-Seba, J. Phys. A., 2000)"

3 comments:

Yes, the renormalization group and the central limit theorem are closely related. A version of the central limit theorem can be arrived at by RG-like methods: http://web.math.princeton.edu/facultypapers/Sinai/KolmogorovLec07.pdf

Similarly, universal phenomena in chaos such as the Feigenbaum number can be deduced by RG-like methods:https://arxiv.org/abs/1210.2262

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About Me

I have fun at work trying to use quantum many-body theory to understand electronic properties of complex materials.
I am married to the lovely Robin and have two adult children and a dog, Priya (in the photo). I also write an even more personal blog Soli Deo Gloria [thoughts on theology, science, and culture]

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